P-ADIC GEOMETRY

Total Page:16

File Type:pdf, Size:1020Kb

P-ADIC GEOMETRY P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (899–934) p-ADIC GEOMETRY P S Abstract We discuss recent developments in p-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for “compact p-adic manifolds” over new period maps on moduli spaces of abelian varieties to applications to the local and global Langlands conjectures, and the con- struction of “universal” p-adic cohomology theories. We finish with some specula- tions on how a theory that combines all primes p, including the archimedean prime, might look like. 1 Introduction In this survey paper, we want to give an introduction to the world of ideas which the author has explored in the past few years, and indicate some possible future directions. The two general themes that dominate this work are the cohomology of algebraic vari- eties, and the local and global Langlands correspondences. These two topics are classi- cally intertwined ever since the cohomology of the moduli space of elliptic curves and more general Shimura varieties has been used for the construction of Langlands corre- spondences. Most of our work so far is over p-adic fields, where we have established analogues of the basic results of Hodge theory for “compact p-adic manifolds”, have constructed a “universal” p-adic cohomology theory, and have made progress towards establishing the local Langlands correspondence for a general p-adic reductive group by using a theory of p-adic shtukas, and we will recall these results below. However, here we wish to relay another, deeper, relation between the cohomology of algebraic varieties and the structures underlying the Langlands corresondence, a re- lation that pertains not to the cohomology of specific algebraic varieties, but to the very notion of what “the” cohomology of an algebraic variety is. Classically, the study of the latter is the paradigm of “motives” envisioned by Grothendieck; however, that vi- sion has still only been partially realized, by Voevodsky [2000], and others. Basically, Grothendieck’s idea was to find the “universal” cohomology as the universal solution to a few basic axioms; in order to see that this has the desired properties, one however needs to know the existence of “enough” algebraic cycles as encoded in the standard conjec- tures, and more generally the Hodge and Tate conjectures. However, little progress has been made on these questions. We propose to approach the subject from the other side MSC2010: primary 14G22; secondary 11S37, 11R39, 11F80, 14G20. Keywords: Perfectoid spaces, rigid-analytic geometry, p-adic Hodge theory, Shimura varieties, Langlands program, shtukas, twistors. 899 900 PETER SCHOLZE and construct an explicit cohomology theory that practically behaves like a universal cohomology theory (so that, for example, it specializes to all other known cohomology theories); whether or not it is universal in the technical sense of being the universal solution to certain axioms will then be a secondary question. This deeper relation builds on the realization of Drinfeld [1980], that in the function field case, at the heart of the Langlands correspondence lie moduli spaces of shtukas. Anderson [1986], Goss [1996], and others have since studied the notion of t-motives, which is a special kind of shtuka, and is a remarkable function field analogue of mo- tives, however without any relation to the cohomology of algebraic varieties. What we are proposing here is that, despite extreme difficulties in making sense of this, there should exist a theory of shtukas in the number field case, and that the cohomology of an algebraic variety, i.e. a motive, should be an example of such a shtuka. This picture has been essentially fully realized in the p-adic case. In the first sections of this survey, we will explain these results in the p-adic case; towards the end, we will then speculate on how the full picture over Spec Z should look like, and give some evidence that this is a reasonable picture. Acknowledgments. This survey was written in relation to the author’s lecture at the ICM 2018. Over the past years, I have benefitted tremendously from discussions with many mathematicians, including Bhargav Bhatt, Ana Caraiani, Gerd Faltings, Laurent Fargues, Ofer Gabber, Eugen Hellmann, Lars Hesselholt, Kiran Kedlaya, Mark Kisin, Arthur-César le Bras, Akhil Mathew, Matthew Morrow, Michael Rapoport, Richard Taylor, and many others. It is a great pleasure to thank all of them for sharing their insights, and more generally the whole mathematical community for being so generous, inviting, supportive, challenging, and curious. I feel very glad and honoured to be part of this adventure. 2 p-adic Hodge theory Let us fix a complete algebraically closed extension C of Qp. The analogue of a com- pact complex manifold in this setting is a proper smooth rigid-analytic variety X over C .1 Basic examples include the analytification of proper smooth algebraic varieties over C , or the generic fibres of proper smooth formal schemes over the ring of integers OC of C . More exotic examples with no direct relation to algebraic varieties are given by the Hopf surfaces X = ((A2 )rig (0; 0) )/qZ ; C n f g where q C is an element with 0 < q < 1 acting via diagonal multiplication on A2. 2 j j Recall that their complex analogues are non-Kähler, as follows from the asymmetry 0 1 1 in Hodge numbers, H (X; ΩX ) = 0 while H (X; OX ) = C . On the other hand, some other non-Kähler manifolds such as the Iwasawa manifolds do not have a p-adic analogue. 1Rigid-analytic varieties were first defined by Tate [1971], and alternative and more general foundations have been proposed by various authors, including Raynaud [1974], Berkovich [1993], Fujiwara and Kato [2006], and Huber [1996]. We have found Huber’s setup to be the most natural, and consequently we will often implicitly regard all schemes, formal schemes and rigid spaces that appear in the following as adic spaces in the sense of Huber; his category naturally contains all of these categories as full subcategories. p-ADIC GEOMETRY 901 The basic cohomological invariants of a compact complex manifold also exist in this i setting. The analogue of singular cohomology is étale cohomology H´et(X; Z`), which is defined for any prime `, including ` = p. If ` p, it follows for example from ¤ work of Huber [1996, Proposition 0.5.3], that this is a finitely generated Z`-module. For ` = p, this is also true by Scholze [2013a, Theorem 1.1], but the argument is significantly harder. i Moreover, one has de Rham cohomology groups HdR(X/C ) and Hodge cohomol- i j ogy groups H (X; ΩX ), exactly as for compact complex manifolds. These are finite- i dimensional by a theorem of Kiehl [1967]. By the definition of HdR(X/C ) as the hy- percohomology of the de Rham complex, one finds an E1-spectral sequence Eij = H j (X; Ωi ) H i+j (X/C ) 1 X ) dR called the Hodge-to-de Rham spectral sequence. In complex geometry, a basic conse- quence of Hodge theory is that this spectral sequence degenerates at E1 if X admits a Kähler metric. This assumption is not necessary in p-adic geometry: Theorem 2.1 (Scholze [2013a, Corollary 1.8], Bhatt, Morrow, and Scholze [2016, The- orem 13.12]). For any proper smooth rigid-analytic space X over C , the Hodge-to- de Rham spectral sequence Eij = H j (X; Ωi ) H i+j (X/C ) 1 X ) dR degenerates at E1. Moreover, for all i 0, i i j j i i X dimC H (X; ΩX ) = dimC HdR(X/C ) = dimQp H´et(X; Qp) : j =0 Fortunately, the Hodge-to-de Rham spectral sequence does degenerate for the Hopf surface – and the examples of nondegeneration such as the Iwasawa manifolds do not have p-adic analogues. Over the complex numbers, the analogue of the equality i i dimC HdR(X/C ) = dimQp H´et(X; Qp) follows from the comparison isomorphism between singular and de Rham cohomology. In the p-adic case, the situation is slightly more complicated, and the comparison iso- morphism only exists after extending scalars to Fontaine’s field of p-adic periods BdR. If X is only defined over C , it is nontrivial to formulate the correct statement, as there is no natural map C BdR along which one can extend scalars; the correct statement ! is Theorem 6.3 below. There is however a different way to obtain the desired equality of dimensions. This relies on the Hodge–Tate spectral sequence, a form of which is implicit in Faltings’s proof of the Hodge–Tate decomposition, Faltings [1988]. Theorem 2.2 (Scholze [2013b, Theorem 3.20], Bhatt, Morrow, and Scholze [2016, The- orem 13.12]). For any proper smooth rigid-analytic space X over C , there is a Hodge– Tate spectral sequence ij i j i+j E = H (X; Ω )( j ) H (X; Zp) Z C 2 X ) ´et ˝ p 902 PETER SCHOLZE that degenerates at E2. Here, ( j ) denotes a Tate twist, which becomes important when one wants to make everything Galois-equivariant. Note that the Hodge cohomology groups appear in the other order than in the Hodge-to-de Rham spectral sequence. Remark 2.3. If X is the base change of a proper smooth rigid space defined over a discretely valued field K C , then everything in sight carries a Galois action, and it follows from the results of Tate [1967], that there is a unique Galois-equivariant splitting of the abutment filtration, leading to a Galois-equivariant isomorphism i i i j j H (X; Qp) Q C M H (X; Ω )( j ) ; ´et ˝ p Š X j =0 answering a question of Tate [ibid., Section 4.1, Remark].
Recommended publications
  • Marco D'addezio
    Marco D’Addezio Max Planck Institute for Mathematics Vivatsgasse 7 Office 221 53111 Bonn, Germany email: [email protected] url: https://guests.mpim-bonn.mpg.de/daddezio Born: January 07, 1992—Rome, Italy Nationality: Italian Current position 2019-2021 Postdoctoral Fellow, Max Planck Institute for Mathematics, Bonn, Germany; Mentor: Prof. Peter Scholze. Previous position 2019 Postdoctoral Fellow, Free University Berlin, Germany; Mentor: Dr. Simon Pepin Lehalleur; Project title: Overconvergent F -isocrystals and rational points. Areas of specialization Algebraic Geometry, Number Theory. Education 2016-2019 Ph.D. in Mathematics, Free University Berlin, Germany; Supervisors: Prof. Hélène Esnault and Prof. Vasudevan Srinivas; Thesis title: Monodromy groups in positive characteristic. 2015-2016 Master 2 - Analyse, Arithmétique et Géométrie, University Paris Sud, France. 2014-2015 Master 1 - Mathematiques Fondamentales et Appliquées, University Paris Sud, France. 2011-2014 Bachelor in Mathematics, University of Pisa, Italy. Grants & awards 2020 Ernst Reuter Prize for the PhD thesis. 2019-2021 Max Planck Institute for Mathematics Postdoctoral Fellowship. 1 2019 German Research Foundation Postdoctoral Fellowship. 2016-2019 Einstein Foundation Fellowship. 2014-2016 Paris Saclay Scholarship. 2011-2014 Istituto Nazionale di Alta Matematica Scholarship. Articles & talks Accepted articles 2021 M. D’Addezio, H. Esnault, On the universal extensions in Tannakian categories, arXiv: 2009.14170, to appear in Int. Math. Res. Not. 2021 M. D’Addezio, On the semi-simplicity conjecture for Qab, arXiv:1805.11071, to appear in Math. Res. Lett. 2020 M. D’Addezio, The monodromy groups of lisse sheaves and overconvergent F -isocrystals, Sel. Math. (New Ser.) 26 (2020). Preprints 2021 M. D’Addezio, Slopes of F-isocrystals over abelian varieties, https://arxiv.org/abs/2101.
    [Show full text]
  • An Important Visitor in Vancouver and Ottawa
    PERSPECTIVES An important visitor in Vancouver and Ottawa Federal research minister Anja Karliczek visits German-Canadian cooperation projects of the Max Planck Society Interested in quantum technology: Federal Research Minister, Anja Karliczek, learned about the Max Planck Center in Vancouver from Max Planck Director Bernhard Keimer (left) and his Canadian colleague Andrea Damascelli. Committee on Education, Research, and Technology Assessment. In Van- couver, the delegation gained insights into research projects at the Max Planck- UBC-U Tokyo Center for Quantum Materials. The Center is home to a close collaboration between several Max Planck Institutes, the University of British Columbia in Vancouver and the University of Tokyo. Two of its Co- Directors, Bernhard Keimer of the Max Planck Institute for Solid State Research and his Canadian colleague Andrea Damascelli, presented the minister with an overview of initial successes that have emerged from collaboration in the area of high-temperature superconduc- tors. On the visit to the Max Planck Uni- versity of Ottawa Centre for Extreme As part of a trip to Canada, the German make key contributions to the explo- and Quantum Photonics, researchers Federal Minister for Education and Re- ration of quantum technologies and explained to the delegation how they search, Anja Karliczek, also paid a visit the international exchange of scien- are developing high-intensity laser to the Max Planck Centers in Vancouver tists,” said the minister, who was accom- sources with a view to optimizing man- and Ottawa. “The Max Planck Centers panied by members of the Bundestag ufacturing processes in the future. Fields Medal for Peter Scholze The new Director at the Max Planck Institute for Mathematics is awarded the highest distinction in his field The Fields Medal is considered the Nobel Prize of mathematics, Exceptional talent: Peter Scholze, and this year the International Mathematical Union chose to a professor at the University of Bonn and Director at the Max Planck award it to Peter Scholze.
    [Show full text]
  • Peter Scholze Awarded the Fields Medal
    Peter Scholze awarded the Fields medal Ulrich Görtz Bonn, October 1, 2018 . Most important research prize in mathematics John Charles Fields Since 1936, 59 medals awarded. Age limit: 40 years . Most important research prize in mathematics . Most important research prize in mathematics . Most important research prize in mathematics . Goal of this talk Some impression of the area, provide context for non-experts. Urbano Monte’s map of the earth, 1587 David Rumsey Map Collection CC-BY-NC-SA 3.0 . Goal of this talk Some impression of the area, provide context for non-experts. Urbano Monte’s map of the earth, 1587 David Rumsey Map Collection CC-BY-NC-SA 3.0 . Goal of this talk Some impression of the area, provide context for non-experts. Urbano Monte’s map of the earth, 1587 David Rumsey Map Collection CC-BY-NC-SA 3.0 . Goal of this talk mod p fields, Archimedean e. g. Fp((t)) fields, e. g. R, C p-adic fields, e. g. Qp Algebraic number fields, e. g. Q . Do solutions exist? Are there only finitely many solutions? Can we count them? Can we write them down explicitly? If there are infinitely many solutions, does the set of solutions have a (geometric) structure? Solving equations Important problem in mathematics: Understand set of solutions of an equation. Solving equations Important problem in mathematics: Understand set of solutions of an equation. Do solutions exist? Are there only finitely many solutions? Can we count them? Can we write them down explicitly? If there are infinitely many solutions, does the set of solutions have a (geometric) structure? .
    [Show full text]
  • Linking Together Members of the Mathematical Carlos ­Rocha, University of Lisbon; Jean Taylor, Cour- Community from the US and Abroad
    NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Features Epimorphism Theorem Prime Numbers Interview J.-P. Bourguignon Societies European Physical Society Research Centres ESI Vienna December 2013 Issue 90 ISSN 1027-488X S E European M M Mathematical E S Society Cover photo: Jean-François Dars Mathematics and Computer Science from EDP Sciences www.esaim-cocv.org www.mmnp-journal.org www.rairo-ro.org www.esaim-m2an.org www.esaim-ps.org www.rairo-ita.org Contents Editorial Team European Editor-in-Chief Ulf Persson Matematiska Vetenskaper Lucia Di Vizio Chalmers tekniska högskola Université de Versailles- S-412 96 Göteborg, Sweden St Quentin e-mail: [email protected] Mathematical Laboratoire de Mathématiques 45 avenue des États-Unis Zdzisław Pogoda 78035 Versailles cedex, France Institute of Mathematicsr e-mail: [email protected] Jagiellonian University Society ul. prof. Stanisława Copy Editor Łojasiewicza 30-348 Kraków, Poland Chris Nunn e-mail: [email protected] Newsletter No. 90, December 2013 119 St Michaels Road, Aldershot, GU12 4JW, UK Themistocles M. Rassias Editorial: Meetings of Presidents – S. Huggett ............................ 3 e-mail: [email protected] (Problem Corner) Department of Mathematics A New Cover for the Newsletter – The Editorial Board ................. 5 Editors National Technical University Jean-Pierre Bourguignon: New President of the ERC .................. 8 of Athens, Zografou Campus Mariolina Bartolini Bussi GR-15780 Athens, Greece Peter Scholze to Receive 2013 Sastra Ramanujan Prize – K. Alladi 9 (Math. Education) e-mail: [email protected] DESU – Universitá di Modena e European Level Organisations for Women Mathematicians – Reggio Emilia Volker R. Remmert C. Series ............................................................................... 11 Via Allegri, 9 (History of Mathematics) Forty Years of the Epimorphism Theorem – I-42121 Reggio Emilia, Italy IZWT, Wuppertal University [email protected] D-42119 Wuppertal, Germany P.
    [Show full text]
  • Arxiv:1711.06456V4 [Math.AG] 1 Dec 2018
    PURITY FOR THE BRAUER GROUP KĘSTUTIS ČESNAVIČIUS Abstract. A purity conjecture due to Grothendieck and Auslander–Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension ě 2. The combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic p0,pq. We establish the remaining cases by using the tilting equivalence for perfectoid rings. To reduce to perfectoids, we control the change of the Brauer group of the punctured spectrum of a local ring when passing to a finite flat cover. 1. The purity conjecture of Grothendieck and Auslander–Goldman .............. 1 Acknowledgements ................................... ................................ 3 2. Passage to a finite flat cover ................................................... .... 3 3. Passage to the completion ................................................... ....... 6 4. The p-primary Brauer group in the perfectoid case .............................. 7 5. Passage to perfect or perfectoid towers ........................................... 11 6. Global conclusions ................................................... ............... 13 Appendix A. Fields of dimension ď 1 ................................................ 15 References ................................................... ........................... 16 1. The purity conjecture of Grothendieck and Auslander–Goldman Grothendieck predicted in [Gro68b, §6] that the cohomological Brauer group of a regular scheme X is insensitive to removing a closed subscheme Z Ă X of codimension ě 2. This purity conjecture is known in many cases (as we discuss in detail below), for instance, for cohomology classes of order 2 invertible on X, and its codimension requirement is necessary: the Brauer group of AC does not agree with that of the complement of the coordinate axes (see [DF84, Rem. 3]). In this paper, we finish the remaining cases, that is, we complete the proof of the following theorem.
    [Show full text]
  • An Integral Model of the Perfectoid Modular Curve
    An integral model of the perfectoid modular curve Juan Esteban Rodr´ıguez Camargo Abstract We construct an integral model of the perfectoid modular curve. Studying this object, we prove some vanishing results for the coherent cohomology at perfectoid level. We use a local duality theorem at finite level to compute duals for the coherent cohomology of the integral perfectoid curve. Specializing to the structural sheaf, we can describe the dual of the completed cohomology as the inverse limit of the integral cusp forms of weight 2 and trace maps. Introduction Throughout this document we fix a prime number p, Cp the p-adic completion of an algebraic closure of Qp, and {ζm}m∈N ⊂ Cp a compatible system of primitive roots of unity. Given a non-archimedean field K we let OK denote its valuation ring. We ˘ let Fp be the residue field of OCp and Zp = W (Fp) ⊂ Cp the ring of Witt vectors. cyc ˘cyc Let Zp and Zp denote the p-adic completions of the p-adic cyclotomic extensions ˘cyc of Zp and Zp in Cp respectively. Let M ≥ 1 be an integer and Γ(M) ⊂ GL2(Z) the principal congruence subgroup of n level M. We fix N ≥ 3 an integer prime to p. For n ≥ 0 we denote by Y (Np )/ Spec Zp the integral modular curve of level Γ(Npn) and X(Npn) its compactification, cf. [KM85]. We denote by X(Npn) the completion of X(Npn) along its special fiber, n and by X (Np ) its analytic generic fiber seen as an adic space over Spa(Qp, Zp), cf.
    [Show full text]
  • Number-Theory Prodigy Among Winners of Coveted Maths Prize Fields Medals Awarded to Researchers in Number Theory, Geometry and Differential Equations
    NEWS IN FOCUS nature means these states are resistant to topological states. But in 2017, Andrei Bernevig, Bernevig and his colleagues also used their change, and thus stable to temperature fluctua- a physicist at Princeton University in New Jersey, method to create a new topological catalogue. tions and physical distortion — features that and Ashvin Vishwanath, at Harvard University His team used the Inorganic Crystal Structure could make them useful in devices. in Cambridge, Massachusetts, separately pio- Database, filtering its 184,270 materials to find Physicists have been investigating one class, neered approaches6,7 that speed up the process. 5,797 “high-quality” topological materials. The known as topological insulators, since the prop- The techniques use algorithms to sort materi- researchers plan to add the ability to check a erty was first seen experimentally in 2D in a thin als automatically into material’s topology, and certain related fea- sheet of mercury telluride4 in 2007 and in 3D in “It’s up to databases on the basis tures, to the popular Bilbao Crystallographic bismuth antimony a year later5. Topological insu- experimentalists of their chemistry and Server. A third group — including Vishwa- lators consist mostly of insulating material, yet to uncover properties that result nath — also found hundreds of topological their surfaces are great conductors. And because new exciting from symmetries in materials. currents on the surface can be controlled using physical their structure. The Experimentalists have their work cut out. magnetic fields, physicists think the materials phenomena.” symmetries can be Researchers will be able to comb the databases could find uses in energy-efficient ‘spintronic’ used to predict how to find new topological materials to explore.
    [Show full text]
  • PETER SCHOLZE to RECEIVE 2013 SASTRA RAMANUJAN PRIZE the 2013 SASTRA Ramanujan Prize Will Be Awarded to Professor Peter Scholze
    PETER SCHOLZE TO RECEIVE 2013 SASTRA RAMANUJAN PRIZE The 2013 SASTRA Ramanujan Prize will be awarded to Professor Peter Scholze of the University of Bonn, Germany. The SASTRA Ramanujan Prize was established in 2005 and is awarded annually for outstanding contributions by young mathematicians to areas influenced by the genius Srinivasa Ramanujan. The age limit for the prize has been set at 32 because Ramanujan achieved so much in his brief life of 32 years. The prize will be awarded in late December at the International Conference on Number Theory and Galois Representations at SASTRA University in Kumbakonam (Ramanujan's hometown) where the prize has been given annually. Professor Scholze has made revolutionary contributions to several areas at the interface of arithmetic algebraic geometry and the theory of automorphic forms. Already in his master's thesis at the University of Bonn, Scholze gave a new proof of the Local Langlands Conjecture for general linear groups. There were two previous approaches to this problem, one by Langlands{Kottwitz, and another by Harris and Taylor. Scholze's new approach was striking for its efficiency and simplicity. Scholze's proof is based on a novel approach to calculate the zeta function of certain Shimura varieties. This work completed in 2010, appeared in two papers in Inventiones Mathematicae in 2013. Scholze has generalized his methods partly in collaboration with Sug Woo Shin to determine the `-adic Galois representations defined by a class of Shimura varieties. These results are contained in two papers published in 2013 in the Journal of the American Mathematical Society. While this work for his master's was groundbreaking, his PhD thesis written under the direction of Professor Michael Rapoport at the University of Bonn was a more mar- velous breakthrough and a step up in terms of originality and insight.
    [Show full text]
  • Public Recognition and Media Coverage of Mathematical Achievements
    Journal of Humanistic Mathematics Volume 9 | Issue 2 July 2019 Public Recognition and Media Coverage of Mathematical Achievements Juan Matías Sepulcre University of Alicante Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Arts and Humanities Commons, and the Mathematics Commons Recommended Citation Sepulcre, J. "Public Recognition and Media Coverage of Mathematical Achievements," Journal of Humanistic Mathematics, Volume 9 Issue 2 (July 2019), pages 93-129. DOI: 10.5642/ jhummath.201902.08 . Available at: https://scholarship.claremont.edu/jhm/vol9/iss2/8 ©2019 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Public Recognition and Media Coverage of Mathematical Achievements Juan Matías Sepulcre Department of Mathematics, University of Alicante, Alicante, SPAIN [email protected] Synopsis This report aims to convince readers that there are clear indications that society is increasingly taking a greater interest in science and particularly in mathemat- ics, and thus society in general has come to recognise, through different awards, privileges, and distinctions, the work of many mathematicians.
    [Show full text]
  • Elliptic Curves, Random Matrices and Orbital Integrals
    ELLIPTIC CURVES, RANDOM MATRICES AND ORBITAL INTEGRALS JEFFREY D. ACHTER AND JULIA GORDON ABSTRACT. An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a product formula, in terms of congruence considerations involving that polynomial, for the size of such an isogeny class (over a finite prime field). In this paper, we give a new, transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality. This answers a question posed by N. Katz in [13, Remark 8.7] and extends Gekeler’s work to ordinary elliptic curves over arbitrary finite fields. 1. INTRODUCTION The isogeny class of an elliptic curve over a finite field Fp of p elements is determined by its trace of Frobenius;p calculating the size of such an isogeny class is a classical problem. Fix a number a with jaj ≤ 2 p, and let I(a, p) be the set of all elliptic curves over Fp with trace of Frobenius a. Further suppose that p - a, so that the isogeny class is ordinary. Gekeler proposes ([9]; see also [13]) a random matrix model to compute the size of I(a, p). For each rational prime ` 6= p, let # fg 2 GL (Z/`n) : tr(g) ≡ a mod `n, det(g) ≡ p mod `ng ( ) = 2 (1.1) n` a, p lim n n . n!¥ # SL2(Z/` )/` For ` = p, let # fg 2 M (Z/pn) : tr(g) ≡ a mod pn, det(g) ≡ p mod png ( ) = 2 (1.2) np a, p lim n n .
    [Show full text]
  • Lectures on Analytic Geometry Peter Scholze (All Results Joint with Dustin
    Lectures on Analytic Geometry Peter Scholze (all results joint with Dustin Clausen) Contents Analytic Geometry 5 Preface 5 1. Lecture I: Introduction 6 2. Lecture II: Solid modules 11 3. Lecture III: Condensed R-vector spaces 16 4. Lecture IV: M-complete condensed R-vector spaces 20 Appendix to Lecture IV: Quasiseparated condensed sets 26 + 5. Lecture V: Entropy and a real BdR 28 6. Lecture VI: Statement of main result 33 Appendix to Lecture VI: Recollections on analytic rings 39 7. Lecture VII: Z((T ))>r is a principal ideal domain 42 8. Lecture VIII: Reduction to \Banach spaces" 47 Appendix to Lecture VIII: Completions of normed abelian groups 54 Appendix to Lecture VIII: Derived inverse limits 56 9. Lecture IX: End of proof 57 Appendix to Lecture IX: Some normed homological algebra 65 10. Lecture X: Some computations with liquid modules 69 11. Lecture XI: Towards localization 73 12. Lecture XII: Localizations 79 Appendix to Lecture XII: Topological invariance of analytic ring structures 86 Appendix to Lecture XII: Frobenius 89 Appendix to Lecture XII: Normalizations of analytic animated rings 93 13. Lecture XIII: Analytic spaces 95 14. Lecture XIV: Varia 103 Bibliography 109 3 Analytic Geometry Preface These are lectures notes for a course on analytic geometry taught in the winter term 2019/20 at the University of Bonn. The material presented is part of joint work with Dustin Clausen. The goal of this course is to use the formalism of analytic rings as defined in the course on condensed mathematics to define a category of analytic spaces that contains (for example) adic spaces and complex-analytic spaces, and to adapt the basics of algebraic geometry to this context; in particular, the theory of quasicoherent sheaves.
    [Show full text]
  • The Perfectoid Concept: Test Case for an Absent Theory
    The Perfectoid Concept: Test Case for an Absent Theory MICHAEL HARRIS Department of Mathematics Columbia University Perfectoid prologue It's not often that contemporary mathematics provides such a clear-cut example of concept formation as the one I am about to present: Peter Scholze's introduction of the new notion of perfectoid space. The 23-year old Scholze first unveiled the concept in the spring of 2011 in a conference talk at the Institute for Advanced Study in Princeton. I know because I was there. This was soon followed by an extended visit to the Institut des Hautes Études Scientifiques (IHES) at Bûres- sur-Yvette, outside Paris — I was there too. Scholze's six-lecture series culminated with a spectacular application of the new method, already announced in Princeton, to an outstanding problem left over from the days when the IHES was the destination of pilgrims come to hear Alexander Grothendieck, and later Pierre Deligne, report on the creation of the new geometries of their day. Scholze's exceptionally clear lecture notes were read in mathematics departments around the world within days of his lecture — not passed hand-to-hand as in Grothendieck's day — and the videos of his talks were immediately made available on the IHES website. Meanwhile, more killer apps followed in rapid succession in a series of papers written by Scholze, sometimes in collaboration with other mathematicians under 30 (or just slightly older), often alone. By the time he reached the age of 24, high-level conference invitations to talk about the uses of perfectoid spaces (I was at a number of those too) had enshrined Scholze as one of the youngest elder statesmen ever of arithmetic geometry, the branch of mathematics where number theory meets algebraic geometry.) Two years later, a week-long meeting in 2014 on Perfectoid Spaces and Their Applications at the Mathematical Sciences Research Institute in Berkeley broke all attendance records for "Hot Topics" conferences.
    [Show full text]